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A coupled discrete/continuous method for computing
lattices. Application to a masonry-like structure
Mohammad Hammoud, Karam Sab, Denis Duhamel
To cite this version:
A Coupled Discrete/Continuous Method for Computing
Lattices. Application to a Masonry-Like Structure
Mohammad Hammouda,∗, Karam Saba, Denis Duhamela
a
Universit´e Paris-Est, UR Navier,Ecole des Ponts ParisTech, 6 et 8 Avenue Blaise Pascal, Cit´e Descartes, Champs sur Marne,
77455 Marne La Vall´ee, Cedex 2, France
Abstract
This paper presents a coupled Discrete/Continuous method for computing lattices and its application to a masonry-like structure. This method was proposed and validated in the case of a one dimensionnal (1D) railway track example presented in (Hammoud et al. (2009)). We study here a 2D model which consists of a regular lattice of square rigid grains interacting by their elastic interfaces. Two models have been developed, a discrete one and a continuous one. In the discrete model, the grains which form the lattice are considered as rigid bodies connected by elastic interfaces (elastic thin joints). In other words, the lattice is seen as a “skeleton” in which the interactions between the rigid grains are represented by forces and moments which depend on their relative displacements and rotations. The continuous model is based on the homogenization of the discrete model (Cecchi & Sab (2009)). Considering the case of singularities within the lattice (a crack for example), we develop a coupled model which uses the discrete model in
∗Corresponding Author, actually at Institut Pprime, ENSMA. Tel: +33 5 49 49 82 26.
singular zones (zones where the discrete model cannot be homogenized), and the continuous model elsewhere. A criterion of coupling is developed and applied at the interface between the discrete and the continuum zones. It verifies the convergence of the coupled solution to the discrete one and limits the size of the discrete zone. A good agreement between the full discrete model and the coupled one is obtained. By using the coupled model, an important reduction in the number of degrees of freedom and in the computation time compared to that needed for the discrete approach, is observed.
Key words: Discrete, Finite Element, Homogenization, Masonry, Interface,
1. Introduction
The aim of this paper is to present an application to 2D masonry pannels of the coupled method between discrete and continuum media already
proposed and validated in the case of a 1D structure by (Hammoud et al. (2009)). The 1D model consisted of a beam resting on an elastic springs. The deflection of the beam (as well as the nodal parameters) was calculated by using two approaches; a discrete approach and a macroscopic approach deduced from the discrete one. A comparison between the response of the system obtained by using these approaches showed the cases where the macroscopic approach cannot replace the discrete one. This difference leaded us to apply a Discrete/Continuum coupling method. A criterion of coupling was developped. In the coupled approach, the macroscopic scale was the intial scale computation. A local discrete computation was done on each macroscopic element. A comparion was done between the nodal
parameters computed by the local discrete method and the continuum one. If a strong difference was observed, a refinement of the computation scale was done. This procedure of refinement was necessary in the zone of singularities.
and moments which depend on their relative displacements and rotations. The second model is a continuous one based on the homogenization of the discrete model.
Many coupled model between the discrete and continuous media are developed. See among others the works of Broughton et al. (1999), Curtin et al. (2003), Wagner et al. (2003), Fish et al. (2004), Xiao & Belytschko (2004), Ricci et al (2005), Frangin et al. (2006) and Klein et al. (2006). In these works, the domain is decomposed into a discrete zone, a continuous zone and an interface zone between the discrete and continuous zones. The interface zone can be a bridging or a handshaking zone where the two descriptions of material exist. Thus, the problem of how to partition energy within the overlap zone is important. For the sake of brevity of the text, the litterature review of these coupled models has been omitted. An exhaustive litterature review has been given in Hammoud et al. (2009). In our model, the handshaking zone is replaced by an interface and then the DoFs of the discrete zone are linked to the DoFs of the continuous zone by calculating the interaction rigidity matrix. The total energy of the domain is written as follows :
Etotal = ED+ EC+ EC-D (1)
where ED and EC are the elastic energies of the discrete and the continuum
zones, respectively. EC-D is the energy of the interaction between the
discrete element (DE) and the finite element (FE) of these zones.
require the calculation of the energy and avoids the problem of how to partition this energy between the discrete and continuum zones. We will
calculate the global rigidity matrices (discrete (KD), continuous (KC) and
interaction (KC-D)) and then solve a linear system written as follows :
KC 0 0 KD + KC-D | {z } Ktotal U C UD = Ftotal (2)
In this present research, at first, we present the 2D masonry model. Secondly, we develop the discrete and the continuous models used to calculate the behavior of the masonry pannel. The continuous boundary value problem is solved by using the Finite Element Method. We
2. The discrete model
The 2D model consists of a regular lattice of square rigid grains interacting by their elastic interfaces (see figure 1).
Figure 1 is approximately here
The in-plane motion of the grain can is described by two displacements and one rotation at the center.
The geometry of the lattice is described hereafter. The position of the
center of grain Bi,j, yi,j, in the Euclidean space is formulated as follows :
yi,j = iae1+ jae2 (3)
e1, e2, e3 is an orthonormal base.
So the displacement of the Bi,j grain is an in plane rigid body motion :
u(y) = ui,j+ ωi,j
× (y − yi,j), ∀ y ∈ Bi,j (4)
where
ui,j = ui,j1 e1 + ui,j2 e2 and ωi,j = ωi,j3 e3 (5)
If the mortar joint is modeled as an elastic interface, then the constitutive law is a linear relation between the tractions on the block surfaces and the jump of the displacement :
t= σ n = K.d on S (6)
Here, σ is the stress tensor, n is the normal to the interface S and d is the displacement jump at S. For isotropic mortar, the elastic interface stiffness tensor K is given as:
K= 1
e µ
M
where λM and µM are the Lam´e constants of the mortar and e is the
thickness of the real joint.
The elastic strain energy associated to the interface S is :
W = 1
2 Z
S
d.(K.d)dS (8)
Note that each grain has four neighbours that mean four interfaces in which two are horizontal and two are vertical, as shown in figure 2. The vectors
C+M1 and C−M1 are given by :
C+M 1 = − a 2e1+ ye2 C−M 1 = a 2e1+ ye2 (9)
Figure 2 is approximately here
So the displacement of a point located on the vertical interface is written as follows :
u+(M1) = u(C+) + ω+× C+M1
u−(M
1) = u(C−) + ω−× C−M1
(10) Thus, the displacement jump at S can be written as :
d = u+(M 1) − u−(M1) = d1e1+ d2e2 = (u+ − u−+ (ω−− ω+) y) e 1+ v+ − v−− (ω−+ ω+)a 2 e2 (11)
Let U be the vector of displacement and rotation of two neighbouring
grains : U = [u+v+ω+u−v−ω−]T. Then, the elastic strain energy
associated to the vertical interface takes the following form :
By using the relationship (8), the value of the elastic strain energy is
calculated. So, from (12), we extract the form of the vertical stiffness tensor as follows : K′a ev 0 0 − K′a ev 0 0 0 K ′′a ev − K′′a√2 4ev 0 − K′′a ev − K′′a√2 4ev 0 −K ′′a√2 4ev (K′+ 3K′′)a 24ev 0 K′′a√2 4ev (−K′ + 3K′′)a 24ev −K ′a ev 0 0 K′a ev 0 0 0 −K ′′a ev K′′a√2 4ev 0 K′′a ev K′′a√2 4ev 0 −K ′′a√2 4ev (−K′+ 3K′′)a 24ev 0 K′′a√2 4ev (K′ + 3K′′)a 24ev (13)
Similarly, the form of the horizontal stiffness tensor Khorizontal can be found.
Hence, the vector of all in-plane degrees of freedom of the structure is calculated by solving the following linear system :
KU= F (14)
in which U = [u1v1ω1...uNvNωN]
T
is the vector of all in-plane degrees of freedom of the structure under consideration and
F= [f1t1m1...fNtNmN]T is the vector of all in-plane elastic actions. K
3. The continuum model
The homogenization of periodic discrete materials has been previously presented in (Pradel & Sab (1998a), Pradel & Sab (1998b), Florence & Sab (2005) and Florence & Sab (2006)), for example. The geometry will be discretized by using the Finite Element Method. As mentioned above, the implementation of the homogenized model will be done with a Matlab code in order to couple later, a continuum zone to a discrete one.
Let us consider the static case of the elastic behavior of the domain. The equilibrium equation is written as :
∇ σ + b = 0 (15)
where ∇ is the divergence operator, σ is the Cauchy stress tensor and b the external load applied on the domain. The stress-strain relationship is given by :
σ = C : ǫ (16)
Where C is the homogenized elastic tensor and ǫ is the strain tensor. Using a weak variational formulation, the equilibrium equation (15) is written as follows :
KU= F (17)
where K is the global stiffness matrix of the domain, U is the global vector of nodal displacements and F is the global vector of external forces applied on the finite element nodes.
In other words, K and F are the assembling of the elementary matrix Ke
C is the homogenized elastic tensor. It is written as follows : C= Ahom 1111 0 0 0 Ahom 2222 0 0 0 Ahom 1212 (18) where Ahom 1111 = K′a eh , A hom 2222 = K′a eh and A hom 1212 = 2K′′a eh .
K′ = λM + 2µM, K′′ = µM and eh is the thickness of the joint between two
grains.
4. Numerical simulations
4.1. Discrete model versus continuous model 4.1.1. Compression test
Let us consider a panel (width L and height H) subjected to compression
actions, supported at its left and right edges with u2(X = 0, X = L) = 0,
fixed at the base u1(Y = 0) = u1(Y = 0) = ω3 = 0 and loaded with a
vertical uniform force applied on the upper edge (see figure 3). In this test, any heterogeneity is introduced in the panel.
Figure 3 is approximately here
In the discrete model, the uniform load is applied on each grain center of the upper edge. In the continuous one, the load is applied at the nodes of the finite element. In figure 4, the nodal displacements of the middle line of
the panel (Y = H
2), u2, are represented. We observe a good match between
Figure 4 is approximately here 4.1.2. Shear test
The case of shear stress is investigated in this part. In the discrete model, the panel is under the following boundary conditions:
grains at the top of the panel: uniform horizontal force,
grains in the left side of the panel u2 = 0 and u1, ω3 are free,
grains in the right side of the panel u2 = 0 and u1, ω3 are free,
blocks at the base of the panel u1 = u2 = 0 and ω3 is free.
In the continuous model, the boundary conditions are the following:
u2(x1 = 0) = u2(x1 = L) = 0, u1(x1 = 0) = u2(x1 = 0) = 0 and a horizontal
uniform load is applied at the side x2 = H (see figure 5). As in the
compression test, the medium is considered homogeneous. Figure 5 is approximately here
By considering the discrete medium at coarse scale and the continuous
model at fine scale, it is obtained that the u1 displacements of the middle
line of the panel don’t match correctly and the relative difference is more than 10% (figure 6). If we refine the coarse scale of the discrete medium, this difference will be negligible as we can observe on the (figure 6).
Figure 6 is approximately here
replace correctly the discrete one when there is no singularities in the structure.
It is clear that the computation time and the number of degrees of freedom (DOFs) in the discrete model are more important than that of the
continuous model. In what follows, in the case of the shear test studied above, a simple comparison (table 1) shows the importance of these two factors: computation time and gain in DOFs.
Table 1 is approximately here 4.2. The coupling model
Now we consider a crack in the panel. Near this crack the medium cannot be homogenized. It is noted that the discrete model can be used to
simulate all the medium, but taking into account the computation time and the number of DOFs, it will be better if we can couple the continuous and discrete models, then the discrete model is used in the cracked zone and the continuous one is used elsewhere.
4.2.1. Principle of the coupling model
The medium is decomposed into two regions. The first one is the
continuum region modeled by finite elements (rectangular with two DoFs by node), the second is the discrete region where the Discrete Element (DE) are the centre of grains (3 DoFs at the center of grains). At the interface between these zones, interpolated DE are used to link the FE of the continuum zone to the DE of the discrete zone (see figure 7).
As mentionned before, by noting ED the elastic energy of the discrete zone,
EC the elastic energy of the continuum one and EC-D the energy of the
interaction between the DE and the FE, the total energy of the coupled medium is given by (1):
The interaction energy between two DEs (- and +) is written as follows :
EI= 1 2 U− U+ T KI U− U+ (19)
EI and KI are the interaction energy and the stiffness matrix of the
interface between two adjacents grains, respectively. U− and U+ are the
vectors of displacements and rotation of the grains (-) and (+), respectively. If we consider a FE modeled by DEs, a relationship between the
displacement of the FE node’s () and the displacement of the DE (◦ created inside the FE) can be established by interpolation, using the shape
functions. By noting [U , V , W ]T the vector of displacements and rotation
of a DE and [u1, v1, u2, v2, u3, v3, u4, v4]
T
the vector of nodal displacements of a FE, the relationship writes :
h U , V , W iT = D h u1, v1, u2, v2, u3, v3, u4, v4 iT (20) D is a interpolation matrix.
It is noted that the discrete displacement at the center of the grain (U) is equal to the finite displacement interpolated in the center of the grain (u(x)): U = u(x). The discrete rotation is also in relation with the finite
displacement by: W = 1
2 grad u(x) − grad
Tu
(x) in which x is the vector
At the same time, each DE located at the edge of the discrete zone (BD) is
connected to an interpolated DE located at the edge of the continuum zone
(BC) by adding half of the interaction energy (19) to the total elastic
energy.
Thus, from these two relationships, a DE located in the discrete zone is linked to a FE in the continuum zone. If we use (20) for the interpolated
DE (U− or U+), then the interaction energy(19) between the DE and the
FE will be a quadratic function of UD and UC.
UD and UC are the global displacements vector of the discrete and
continuum zones respectively. By designing KD and KC, the discrete and
continuum stiffness matrices, the total energy of the medium will be :
Etotal= 1 2 U C UD T KC 0 0 KD + KC-D | {z } Ktotal U C UD (21)
KC-D is the global matrix of interaction which is calculated by the
summation of all elementary interaction matrices between the discrete and continuous zones.
4.2.2. Criterion of coupling
Figure 8 is approximately here
The external loading is computed as follows: Using (20), the displacements at the center of the interpolated DE created in the FE can be calculated. From the interaction energy formulated in (19), we calculate the interaction forces and moments between these two DEs using the relation
(F =Kinterface.[U+, U−]T). All the interaction forces between a DE (•)
and an external interpolated DE (◦) at the edge of FE are computed and assembled to form the external global load applied on the discrete zone included in the FE.
Using the discrete model, we calculate the discrete displacements of the DE
noted as Ud
a. After that, we calculate the difference between the
interpolated continuum displacements in (20), Uc
i at the center of grains
and Ud
a. This difference will be the criterion for coupling. It is formulated
as follows : error = Ud a − Uci Ud a (22)
By noting “TEST ZONE” the FE zone neighbouring the discrete one, we check the criterion (22) on each FE of this zone. In other words, we check if the FEs of the “TEST ZONE” lead to the correct solution (we mean by it the full discrete solution). So if the error (22) is more than 10%, the scale of computation will be that of the discrete model. The size of the discrete zone will increase. In the other case (error less than 10%), the continuum scale of computation is adapted and the size of the discrete zone is
4.2.3. Numerical algorithm
At first, the medium is meshed at a coarse scale by using FE. At the center of the medium, a crack is created by broking the interaction between
interfaces. The cracked zone is modeled by DE. The size of the discrete zone is fixed. After applying a traction load, for example, we simulate the response of the medium. At the interface between the discrete zone and the continuum one, we check the criterion of coupling described before.
Hereafter, a diagram of this algorithm is presented. Figure 9 is approximately here 4.2.4. Cracked wall: discrete model vs coupled model
We consider a panel (width L, height H) with a crack at its center. The cracked zone is modeled by DE and the rest of the panel is modeled by FE as shown in figure 10.
Figure 10 is approximately here
Firstly a complete discrete simulation is done in order to compare the coupled solution to that discrete. Let us consider a panel modeled by 25 × 25 grains. After a traction load, we can observe the crack, by simply representing the position of the center of each grain. We can observe in figure 11 the rotation of grains considered like rigid bodies.
Figure 11 is approximately here
Now, let us consider the coupled simulation. The size of a FE is supposed equal to 8 times the size of a DE. The size of the discrete zone is fixed to 3 × 3 FEs which means 72 DEs. The mesh after loading takes the shape seen in figure 12.
Figure 12 is approximately here
If we compare the Y displacements of the middle line of the panel, we can observe a perfect match between the discrete and coupled solutions. This agreement is illustrated in figure 13.
Figure 13 is approximately here 4.2.5. Gain in time and DoFs
In this paragraph, we underline the advantage of this coupled approach. In the coupled simulation done before, by considering the same dimensions of
the panel (4 × 4 m2), the total number of DoFs is the sum of (72 × 3
discrete DoFs) and (85 × 2 continuous DoFs).
The computation time is estimated to 54 seconds. By a simple comparison (see table 2) between discrete and coupled parameters, we can concluded their importance.
Table 2 is approximately here
The gain in DoFs is evaluated to : GDoFs =
1875
386 = 4.86 and the gain in
computation time is : Gtime=
322
54 = 5.96. These gain factors will be more
By applying the numerical criterion at the interface between discrete and continuum zones, the difference between discrete and continuum solutions is evaluated to 9%. This difference can be minimised if we increase the size of the discrete zone. Thus, the gain in DoFs and computation time will decrease.
5. Conclusion
In this work a 2D coupled model between discrete and continuum media has been performed. The discrete model is based on interaction between rigid bodies by their interfaces. The continuous model is based on the homogenization of the discrete model. Numerical simulations show that the discrete medium is homogeneizable if there is no singularities in the
References
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Concurrent coupling of length scales: methodology and application. Physical Review B 60, p. 2391-2403.
Cecchi, A., Sab, K., (2009). Discrete and continuous models for in plane loaded random elastic brickwork. European Journal of Mechanics -A/Solids, 28, p. 610-625.
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Cecchi, A., Sab, K., (2002a). A multi-parameter homogenization study for modelling elastic masonry. Eur. J. Mech. A/Solids 21, p. 249-268.
Cecchi, A., Sab, K., (2002b). Out of plane model for heterogeneous periodic materails: The case of masonry. Eur. J. Mech. A/Solids 21, p. 249-268. Cluni, F., Gusella, V., (2004). Homogenization of non-periodic masonry
structures. International Journal of Solids and Structures 41, (7), p. 1911-1923.
Curtin, W.A., Miller, R.A., (2003). Atomistic/continuum coupling in computational materials science. Modelling and Simulation in Materials Science and Engineering, 11, R33-R68.
Florence, C., Sab, K., (2006). A rigorous homogenization method for the determination of the overall ultimate strength of periodic discrete media and an application to general hexagonal lattices of beams. Eur. J. Mech. A/Solids 25, (1), p. 72-97.
Florence, C., Sab, K., (2005). Overall ultimate surface of periodic tetrakaidecahedral lattice with non-symmetric material distribution. Journal of Materials Sciences, 40, p. 5883-5892.
Frangin, E., Marin, P., Daudeville, L., (2006). Coupled finite/discrete elements method to analyze localized impact on reinforced concrete structure. In: Proceedings EURO-C.
Hammoud, M., Duhamel, D., Sab, K., (2009). Static and dynamic studies for coupling discrete and continuum media; Application to a simple railway track model. International Journal of Solids and Structures 47, p. 276-290.
Klein, P.A., Zimmerman, J.A., (2006). Coupled atomistic-continuum simulations using arbitrary overlapping domains. Journal of Computational Physics 213, p. 86-116.
Pradel, F., Sab, K., (1998a). Homogenization of discrete media. J.Physique IV 8 (P8), p. 317-324.
Pradel, F., Sab, K., (1998b). Cosserat modelling of periodic lattice structures. C.R. Acad. Sci. Paris II b, 326, p. 699-704.
Sab, K., (1996). Microscopic and macroscopic strains in a dense collection of rigid particles. C. R. Acad. Sci. S´er. II Fasc. B - M´ecanique Physique Chimie Astronomie 322 (10), p. 715-721.
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y
2
y1
e
2e
1+
C
−
C
M
2M
1B
E
D
2 Y Y2 1 Y L L F F F F F F F F F F F F F F H H
Figure 3: Masonry panel (width L and height H) subject to compression actions supported at its left and right edges u2 = 0, fixed at the base loaded with a vertical uniform force
0 0.5 1 1.5 2 2.5 3 3.5 4 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0x 10 −11 X coordinates of nodes in (m)
Y Nodes displacements in (m) of the middle line
continuous model discrete model
2 Y 1 Y 2 Y F F F F F H L F F F L H F F F
Figure 5: Masonry pannel (width L and height H) subject to shear actions simply sup-ported at its left and right edge u2 = 0 and fixed at the base loaded with a horizontal
0 0.5 1 1.5 2 2.5 3 3.5 4 −4 −3 −2 −1 0 1 2 3 4x 10 −13
X coordinates of nodes and centers of grains in (m)
Y Nodal displacements in (m) continuous 41x41
discrete 61x61 discrete 31x31 discrete 21x21
BC BC BC BC BD BI L H
Figure 7: Regular lattice of square grains modeled by a coupling discrete/continuum model; (•) are the DE of the region (BD
), (◦) are the interpolated DE of the (BI) and
+ − + − − − f m f m m f f m m f
Coarse continuum mesh
Creation of the crack within the panel
if e>10%
END Discerte mesh near
Computation of the error between the continuum and discrete solutions
near the cracked zone Solution of the linear system Computation of the global stiffness matrix
Assembling of the elementary matrix the cracked zone
Figure 10: Masonry panel (width L and height H) subject to traction actions, fixed at the base and simply supported at its left and right edges u1(Y = H/2, X = 0) = u1(Y =
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
X coordinates of grains centers in (m)
Y coordinates of grains centers in (m)
0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
X coordinates of FE nodes and centers of bricks in (m)
Deformed mesh after loading
0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
X coordinates of FE nodes and center of grains in (m)
Displacements according Y axis in (cm)
Coupled model Discrete model
Number of nodes Number of DoFs Computation time
Discrete model 625 625 × 3 = 1875 322 seconds
Continuous model 72 72 × 2 = 144 35 seconds
Number of nodes Number of DoFs Computation time
Discrete model 625 DEs 625 × 3 = 1875 322 seconds
Coupled model 85 FEs + 72 DEs 85 × 2 + 72 × 3 = 386 54 seconds
